Measure on gauge invariant symmetric norms

The concept of a gauge invariant symmetric random norm is elaborated in this paper. We introduce norm processes and show that this kind of stochastic processes are closely related to gauge invariant symmetric random norms. We construct a gauge invariant symmetric random norm on the plain. We define two different extensions of these random norms to higher (even infinite) dimensions. We calculate numerically unit spheres of expected norms in two and three dimensions for the constructed random norm.